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POLAR COORDINATES AND
POLAR GRAPHS
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At the end of the lecture, you should be able to: Define the coordinates of points using polar
coordinate system.
Translate the rectangular coordinates to polarcoordinates and vice versa.
Draw graphs of equations in polar coordinateaxis.
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THE POLAR COORDINATE SYSTEM
• There are various types of coordinate system.
• The rectangular system is probably the most
important. In this system a point is located by its
distances from two perpendicular lines.• The second type of coordinate system is called
polar coordinates system.
• It is a coordinate system in which the coordinates
of a point in a plane are its distances from a fixed
point and its direction from a fixed line.
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THE POLAR COORDINATE SYSTEM
The reference frame in the polar coordinate
system is a half-line drawn from some point in theplane.
The half-line is represented by OA.
The point O is called the origin or pole and OA isthe polar axis.
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THE POLAR COORDINATE SYSTEM
The position of any point P in the plane is determinedby the distance OP and the angle AOP.
The segment OP, denoted by r , is referred to as radius
vector; The angle AOP, denoted by θ , is called the vectorial
angle.
The coordinates of P is written as P(r, θ ) or just (r, θ ).
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THE POLAR COORDINATE SYSTEM
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The center of the graph
is called the pole.
Angles are measured
from the positive x axis.
Points arerepresented by a
radius and an angle
To plot the point
First find the angle
Then move out along
the terminal side 5units
polar axis
terminal side
(r,
)
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A negative angle would be measured clockwise
like usual.
To plot a point with
a negative radius,
find the terminal
side of the anglebut then measure
from the pole in
the negative
direction of theterminal side.
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THE POLAR COORDINATE SYSTEM
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Let's plot the following points:
We can see thatunlike in the
rectangular
coordinate system,
there are manyways to list the
same point.
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THE POLAR COORDINATE SYSTEM
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THE POLAR COORDINATE SYSTEM
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RELATIONS BETWEEN RECTANGULAR AND POLAR
COORDINATES
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RELATIONS BETWEEN RECTANGULAR AND POLAR
COORDINATES
• To obtain r and θ in terms of x and y, we use thePythagorean theorem and the tangent function. That is
• But the range of tan-1 is –π/2 < θ < π/2 so the value of
θ from the previous equation will not represent any
point to the left of the y-axis. So the measure of the
angle θ is given by
x
yand y xr tan
222
0tan
0tan
1
1
xif
x
y
xif x
y
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RELATIONS BETWEEN RECTANGULAR AND POLAR
COORDINATES
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RELATIONS BETWEEN RECTANGULAR AND POLAR
COORDINATES
Rectangular Form:
Polar Form:
C By Ax
sincos B A
C r
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RELATIONS BETWEEN RECTANGULAR AND POLAR
COORDINATES
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POLAR EQUATIONS and POLAR GRAPHS
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POLAR EQUATIONS and POLAR GRAPHS
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Graph of r = a
In general, the graph of r = a is
a circle with center at the pole.
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POLAR EQUATIONS and POLAR GRAPHS
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Graph of = a
In general, the graph of = a
is a line through the origin.
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POLAR EQUATIONS and POLAR GRAPHS
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POLAR EQUATIONS and POLAR GRAPHS
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POLAR EQUATIONS and POLAR GRAPHS
• Graph
a. r= 7cosθ
b. r= -4sinθ
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POLAR GRAPHS
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POLAR GRAPHS
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• The heart-shaped graph is a cardioid.
• In general, the graph of any of the polar equationsbelow, with a ≠ 0, is a cardioid.
POLAR GRAPHS
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• The heart-shaped graph is a cardioid.
• In general, the graph of any of the polar equationsbelow, with a ≠ 0, is a cardioid.
POLAR GRAPHS
Th h i di id
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Let each unit be 1/4.
The graph is a cardioid .
θ r
0 1
π/2 0π 1
3π/2 2
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POLAR EQUATIONS and POLAR GRAPHS
•
Graphr = 2 + 2cosθ
θr
0 4
π/2 2
π 03π/2 2
Let each unit be 1/2.
The graph is a cardioid .
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POLAR EQUATIONS and POLAR GRAPHS
•
Graphr = 2 + 2cosθ
θ r
0 4
π/6 3.73
π/3 3
π/2 2
2π/3 1
5π/6 0.27
π 0
Let each unit be 1/2.
θ r
7π/6 0.27
4π/3 1
3π/2 2
5π/3 3
11π/6 3.73
2π 4
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POLAR GRAPHS
The graph is a limacon with a dimple.
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Let each unit be 1.
The graph is a limacon with a dimple.
θ r
0 5
π/2 3
π 1
3π/2 3
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Graphs of Polar Equations
Limacon with an inner loop
θ r
0 6
π/2 2
π -2
3π/2 2
Let each unit be 1.
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Graphs of Polar Equations
4
2
2
4
5
Limacon with an inner loop
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Graphs of Polar Equations
Graph
r = 3 + sinθ
Convex Limacon
Let each unit be 1/2.
θr
0 3
π/2 4
π3
3π/2 2
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POLAR GRAPHS
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Graphs of Polar Equations
Four – leafed rose
2
1
1
2
2 2
A
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Let each unit be 1/2.
Graph r= -2sin(3θ)
6,2
r If
3
2
#
2: petalsof
Interval
2
3,2,
6
5,2,
6,2
:
petalstheof Tip
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POLAR GRAPHS
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POLAR GRAPHS
Lemniscates/ Lemniscate of Bernoulli
r²=-a²sin2θ r²=-a²cos2θ
values of θ that make the right member positive
are excluded:
for r²=-a²sin2θ, the excluded values are
0<θ
<π
/2 andπ
<θ
<3π
/2for r²=-a²cos2θ, the excluded values are
-π/4< θ < π/4 and 3π/4< θ < 5π/4
The graph is a lemniscate
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Let each unit be 1/2.
The excluded values
are π/2< θ < π and3π/2< θ < 2π
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Graphs of Polar Equations
lemniscate
2
2
G h f P l E ti
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Graphs of Polar Equations
• Graph
r² = -9 cos2θ
the excluded values
are -π/4< θ < π/4
and 3π/4< θ < 5π/4
The graph is a lemniscate